I grew up with Riemers.net - very nice tutorial site covering everything form basics (drawing Your first triangle) to advanced techniques (HLSL). In every tutorial You create a cool application, like a flight simulator for example.

I found XNA Development: Tutorials for the masses to be absolutely great. It covers basic things well - sprites, moving them, firing things, scrolling backgrounds, and comes with full source. The other material, on game states and other bits, are equally good.

The other thing I found useful, but definitely at a slightly more advanced level, is Nick Gravelyn's tutorial for Alien Aggressors. It's a full featuring Space Invaders/Galaga clone.
I cannot find it, since his site's been taken down. But it was in pdf form and a great read if anyone can find.